It's a nominalization from the Latin verb argūere, 'show, prove, assert, declare, make clear, reprove, accuse, charge with, blame, censure, denounce as false'. It's not what we mean in English as "arguing"; in Latin it had more to do with persuasion and rhetoric than logic -- logic was just a special variety of rhetoric, and mathematics didn't exist as such outside of Euclid and abacus techniques.
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John LawlerJan 1 '14 at 19:14

@John: I'm supposing that in the actual meaning under consideration here, the sense is extrapolated from a specific value (or case) that "bolsters" an argument.
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FumbleFingersJan 1 '14 at 19:20

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+1 The distinction between parameter and argument is important. A parameter is a thing over which something is parameterised; an argument is a value by which you instantiate the parameterised thing. In a typed setting, the type of a function is a theorem, and the function is a proof; when evaluating the proof, an argument is evidence for a specific type.
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Jon PurdyJan 1 '14 at 19:31

@Jon: I'll take that as further evidence that this particular usage can reasonably be linked to the more familiar argument = reason given in support of a theory (which also spreads in another direction to give an exchange of diverging or opposite views, typically a heated or angry one).
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FumbleFingersJan 1 '14 at 22:38

I feel this answer is in some way partial.
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Edwin AshworthJan 1 '14 at 16:36

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First cite of a use in math, maybe, but it can hardly be the first use: it doesn't define it in any way, so if this really was a new meaning at the time, it would have made no sense to anyone.
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MarthaªJan 1 '14 at 16:37

@Martha: I'm tempted to ask “Is that American for hello?” but you might be one of the apparently few people over there who don't watch Downton. (It was nice to see Shirley MacLaine managing to trump Maggie Smith later.)
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Edwin AshworthJan 1 '14 at 16:44

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Well, the answer pretty much answers the QUESTION (about sense, not about origin as such). The jump between "Argument" in Logic and in Astronomy still seems a bit large but I can accept the argument "token of knowledge, that gives desired result after processing".
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SF.Jan 1 '14 at 17:29

BTW, I feel this question would be more aptly asked in the Programmer's/Software engineering forum.

Who used it first - Mathematics or Linguistics?

Regardless of precedence, the use of the term argument certainly was inherited directly from computer science linguistics, which was both a significant benefactor and beneficiary to the field of linguistics.

An argument is a token that forms part of the syntax tree.

When computational linguists (mathematician Admiral Grace Hopper being a well-known one) design or analyse a computational language, they would construct a syntax tree. And guess what ... the term argument would be part of their consideration, wouldn't it?

Grace Hopper being a pioneer in the field of computational linguistics must have contributed significantly to the prevalent use of the term argument in computer programming, as we are well aware she had also given us the term debugging.

Since she was a mathematician, she would have leaned heavily on the meaning of arguments in Mathematics.

Avram Noam Chomsky introduced Phrase-Structural Analysis into linguistics. According to the Wikipedia article on him, he made major contributions to analytic philosophy, and "His work has influenced fields such as artificial intelligence, cognitive science, computer science, logic, mathematics, music theory and analysis, political science, programming language theory and psychology."

I would liken an argument in a syntax tree to a child node to a predicate node.

As example of arguments, while outmoded "Sauserrean" structured grammarians use the term transitive verbs, phrase-structural linguists use the more flexible term valency of a verb. Where, the valency of a verb is the number of arguments a verb would take, as explained in my (less than precise) argument at The subject is after the verb in this sentence?, where I confuse the use of arguments vs adjuncts.

I am sure the precision of my historical account of linguistics is left to desire, as it merely points to the areas of historical research. However, I believe it is accurate in pointing out that the use of the term argument has direct genealogy from computational linguistics which descended from structural analysis in Linguistics, which in turn cross contributed with Mathematics and Philosophy,